TY - JOUR

T1 - Delay asymptotics and bounds for multitask parallel jobs

AU - Wang, Weina

AU - Harchol-Balter, Mor

AU - Jiang, Haotian

AU - Scheller-Wolf, Alan

AU - Srikant, R.

N1 - Funding Information:
Acknowledgements This work was supported in part by National Science Foundation Grants CPS ECCS-1739189, ECCS-1609370, XPS-1629444, and CMMI-1538204, the US Army Research Office (ARO Grant No. W911NF-16-1-0259), the US Office of Naval Research (ONR Grant No. N00014-15-1-2169), DTRA under the Grant Number HDTRA1-16-0017, and a 2018 Faculty Award from Microsoft. Additionally, Haotian Jiang was supported in part by the Department of Physics at Tsinghua University.
Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2019/4/15

Y1 - 2019/4/15

N2 - We study delay of jobs that consist of multiple parallel tasks, which is a critical performance metric in a wide range of applications such as data file retrieval in coded storage systems and parallel computing. In this problem, each job is completed only when all of its tasks are completed, so the delay of a job is the maximum of the delays of its tasks. Despite the wide attention this problem has received, tight analysis is still largely unknown since analyzing job delay requires characterizing the complicated correlation among task delays, which is hard to do. We first consider an asymptotic regime where the number of servers, n, goes to infinity, and the number of tasks in a job, k(n), is allowed to increase with n. We establish the asymptotic independence of any k(n) queues under the condition k(n)= o(n1 / 4). This greatly generalizes the asymptotic independence type of results in the literature, where asymptotic independence is shown only for a fixed constant number of queues. As a consequence of our independence result, the job delay converges to the maximum of independent task delays. We next consider the non-asymptotic regime. Here, we prove that independence yields a stochastic upper bound on job delay for any n and any k(n) with k(n)≤ n. The key component of our proof is a new technique we develop, called “Poisson oversampling.” Our approach converts the job delay problem into a corresponding balls-and-bins problem. However, in contrast with typical balls-and-bins problems where there is a negative correlation among bins, we prove that our variant exhibits positive correlation.

AB - We study delay of jobs that consist of multiple parallel tasks, which is a critical performance metric in a wide range of applications such as data file retrieval in coded storage systems and parallel computing. In this problem, each job is completed only when all of its tasks are completed, so the delay of a job is the maximum of the delays of its tasks. Despite the wide attention this problem has received, tight analysis is still largely unknown since analyzing job delay requires characterizing the complicated correlation among task delays, which is hard to do. We first consider an asymptotic regime where the number of servers, n, goes to infinity, and the number of tasks in a job, k(n), is allowed to increase with n. We establish the asymptotic independence of any k(n) queues under the condition k(n)= o(n1 / 4). This greatly generalizes the asymptotic independence type of results in the literature, where asymptotic independence is shown only for a fixed constant number of queues. As a consequence of our independence result, the job delay converges to the maximum of independent task delays. We next consider the non-asymptotic regime. Here, we prove that independence yields a stochastic upper bound on job delay for any n and any k(n) with k(n)≤ n. The key component of our proof is a new technique we develop, called “Poisson oversampling.” Our approach converts the job delay problem into a corresponding balls-and-bins problem. However, in contrast with typical balls-and-bins problems where there is a negative correlation among bins, we prove that our variant exhibits positive correlation.

KW - Association of random variables

KW - Asymptotic independence

KW - Large systems

KW - Parallel jobs

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U2 - 10.1007/s11134-018-09597-5

DO - 10.1007/s11134-018-09597-5

M3 - Article

AN - SCOPUS:85060174226

SN - 0257-0130

VL - 91

SP - 207

EP - 239

JO - Queueing Systems

JF - Queueing Systems

IS - 3-4

ER -